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G = S3×A5order 360 = 23·32·5

Direct product of S3 and A5

direct product, non-abelian, not soluble, A-group

Aliases: S3×A5, C3⋊(C2×A5), (C3×A5)⋊1C2, SmallGroup(360,121)

Series: ChiefDerived Lower central Upper central

C1C3S3 — S3×A5
A5C3×A5 — S3×A5
C3×A5 — S3×A5
C1

3C2
15C2
45C2
10C3
20C3
6C5
5C22
45C22
45C22
10S3
15C6
15S3
30S3
30C6
60S3
10C32
6D5
18D5
18C10
6C15
15C23
5A4
5C2×C6
10A4
15D6
15D6
30D6
10C3×S3
10C3×S3
10C3⋊S3
18D10
6D15
6C5×S3
6C3×D5
5C22×S3
15C2×A4
5C3×A4
10S32
6S3×D5
5S3×A4
3C2×A5

Character table of S3×A5

 class 12A2B2C3A3B3C5A5B6A6B10A10B15A15B
 size 131545220401212306036362424
ρ1111111111111111    trivial
ρ21-11-1111111-1-1-111    linear of order 2
ρ32020-12-122-1000-1-1    orthogonal lifted from S3
ρ43-3-113001-5/21+5/2-10-1-5/2-1+5/21+5/21-5/2    orthogonal lifted from C2×A5
ρ533-1-13001+5/21-5/2-101-5/21+5/21-5/21+5/2    orthogonal lifted from A5
ρ633-1-13001-5/21+5/2-101+5/21-5/21+5/21-5/2    orthogonal lifted from A5
ρ73-3-113001+5/21-5/2-10-1+5/2-1-5/21-5/21+5/2    orthogonal lifted from C2×A5
ρ84-400411-1-10-111-1-1    orthogonal lifted from C2×A5
ρ94400411-1-101-1-1-1-1    orthogonal lifted from A5
ρ105-51-15-1-100110000    orthogonal lifted from C2×A5
ρ1155115-1-1001-10000    orthogonal lifted from A5
ρ1260-20-3001-51+51000-1-5/2-1+5/2    orthogonal faithful
ρ1360-20-3001+51-51000-1+5/2-1-5/2    orthogonal faithful
ρ148000-42-1-2-2000011    orthogonal faithful
ρ1510020-5-2100-100000    orthogonal faithful

Permutation representations of S3×A5
On 15 points - transitive group 15T23
Generators in S15
(1 2 3 4 5)(6 7 8 9 10 11 12 13 14 15)
(1 15 10)(2 13 14)(3 7 12)(4 9 6)(5 11 8)

G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15), (1,15,10)(2,13,14)(3,7,12)(4,9,6)(5,11,8)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15), (1,15,10)(2,13,14)(3,7,12)(4,9,6)(5,11,8) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10,11,12,13,14,15)], [(1,15,10),(2,13,14),(3,7,12),(4,9,6),(5,11,8)])

G:=TransitiveGroup(15,23);

On 18 points - transitive group 18T145
Generators in S18
(2 3)(4 5 6 7 8)(9 10 11 12 13 14 15 16 17 18)
(1 11 10)(2 16 7)(3 8 15)(4 13 14)(5 9 12)(6 17 18)

G:=sub<Sym(18)| (2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18), (1,11,10)(2,16,7)(3,8,15)(4,13,14)(5,9,12)(6,17,18)>;

G:=Group( (2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18), (1,11,10)(2,16,7)(3,8,15)(4,13,14)(5,9,12)(6,17,18) );

G=PermutationGroup([(2,3),(4,5,6,7,8),(9,10,11,12,13,14,15,16,17,18)], [(1,11,10),(2,16,7),(3,8,15),(4,13,14),(5,9,12),(6,17,18)])

G:=TransitiveGroup(18,145);

On 30 points - transitive group 30T85
Generators in S30
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)
(1 27 17)(2 14 26)(3 29 13)(4 16 28)(5 25 15)(6 20 24)(7 21 19)(8 12 30)(9 23 11)(10 18 22)

G:=sub<Sym(30)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,27,17)(2,14,26)(3,29,13)(4,16,28)(5,25,15)(6,20,24)(7,21,19)(8,12,30)(9,23,11)(10,18,22)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,27,17)(2,14,26)(3,29,13)(4,16,28)(5,25,15)(6,20,24)(7,21,19)(8,12,30)(9,23,11)(10,18,22) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30)], [(1,27,17),(2,14,26),(3,29,13),(4,16,28),(5,25,15),(6,20,24),(7,21,19),(8,12,30),(9,23,11),(10,18,22)])

G:=TransitiveGroup(30,85);

On 30 points - transitive group 30T94
Generators in S30
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)
(1 26 12)(2 17 25)(3 28 16)(4 11 27)(5 13 20)(6 24 18)(7 22 23)(8 30 21)(9 15 29)(10 19 14)

G:=sub<Sym(30)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,26,12)(2,17,25)(3,28,16)(4,11,27)(5,13,20)(6,24,18)(7,22,23)(8,30,21)(9,15,29)(10,19,14)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,26,12)(2,17,25)(3,28,16)(4,11,27)(5,13,20)(6,24,18)(7,22,23)(8,30,21)(9,15,29)(10,19,14) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30)], [(1,26,12),(2,17,25),(3,28,16),(4,11,27),(5,13,20),(6,24,18),(7,22,23),(8,30,21),(9,15,29),(10,19,14)])

G:=TransitiveGroup(30,94);

On 30 points - transitive group 30T102
Generators in S30
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)
(1 3 29)(4 13 28)(5 20 12)(6 17 19)(7 21 16)(8 10 30)(14 25 27)(15 22 24)

G:=sub<Sym(30)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,3,29)(4,13,28)(5,20,12)(6,17,19)(7,21,16)(8,10,30)(14,25,27)(15,22,24)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,3,29)(4,13,28)(5,20,12)(6,17,19)(7,21,16)(8,10,30)(14,25,27)(15,22,24) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30)], [(1,3,29),(4,13,28),(5,20,12),(6,17,19),(7,21,16),(8,10,30),(14,25,27),(15,22,24)])

G:=TransitiveGroup(30,102);

Polynomial with Galois group S3×A5 over ℚ
actionf(x)Disc(f)
15T23x15+5x14+8x13+7x12+8x11-3x9+7x8-2x7+7x6-2x5-3x4+5x3-2x+1-237·416·4632

Matrix representation of S3×A5 in GL5(𝔽31)

01000
10000
0032921
002103
00100
,
030000
130000
00282129
0029310
003000

G:=sub<GL(5,GF(31))| [0,1,0,0,0,1,0,0,0,0,0,0,3,2,1,0,0,29,10,0,0,0,21,3,0],[0,1,0,0,0,30,30,0,0,0,0,0,28,29,30,0,0,21,3,0,0,0,29,10,0] >;

S3×A5 in GAP, Magma, Sage, TeX

S_3\times A_5
% in TeX

G:=Group("S3xA5");
// GroupNames label

G:=SmallGroup(360,121);
// by ID

G=gap.SmallGroup(360,121);
# by ID

Export

Subgroup lattice of S3×A5 in TeX
Character table of S3×A5 in TeX

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